Question

Factorise the following :
1 x^2-y^2-4yz-4z^2
2 2bc - b^2-c^2+4a^2
3. a^2-b^2-a-b
4 a^2b^2+1-a^2-b^2

Answer 1

$1. {x}^{2} - {y}^{2} - 4yz - 4 {z}^{2} \\ {x}^{2} - ( {y}^{2} + 4yz + 4 {z}^{2} ) \\ {x}^{2} - {(y + 2z)}^{2} \\ (x + y + 2z)( x - y - 2z)\\ 2. \: 2bc - {b}^{2} - {c}^{2} + 4 {a}^{2} \\ 4 {a}^{2} - ( {b}^{2} - 2bc + {c}^{2} ) \\ 4 {a}^{2} - {(b - c)}^{2} \\ (2a + b - c)(2a - b + c) \\ 3. \: {a}^{2} - {b}^{2} - a - b \\ (a + b)(a - b) - (a + b) \\ (a + b)(a - b - 1) \\ 4. \: \: {a}^{2} {b}^{2} + 1 - {a}^{2} - {b}^{2} \\ {a}^{2} {b}^{2} - {b}^{2} - {a}^{2} + 1 \\ {b}^{2} ( {a}^{2} - 1) - 1( {a}^{2} - 1) \\ ( {a}^{2} - 1 )( {b}^{2} - 1 ) \\ (a + 1)(a - 1)(b + 1)(b - 1)$

Answer 2

   

$1. \ \ x^2-y^2-4yz-4z^2 \\ \\ \Rightarrow\ x^2-(y^2+4yz+4z^2) \\ \\ \Rightarrow\ x^2-(y+2z)^2 \\ \\ \Rightarrow\ (x+(y+2z))(x-(y+2z)) \\ \\ \Rightarrow\ (x + y + 2z)(x - y - 2z)$

$2.\ \ 2bc-b^2-c^2+4a^2 \\ \\ \Rightarrow\ 4a^2-b^2+2bc-c^2 \\ \\ \Rightarrow\ 4a^2-(b^2-2bc+c^2) \\ \\ \Rightarrow\ 4a^2-(b-c)^2 \\ \\ \Rightarrow\ (2a+(b-c))(2a-(b-c)) \\ \\ \Rightarrow\ (2a+b-c)(2a-b+c)$

$3. \ \ a^2-b^2-a-b \\ \\ \Rightarrow\ (a^2-b^2)-(a+b) \\ \\ \Rightarrow\ (a+b)(a-b)-(a+b) \\ \\ \Rightarrow\ (a+b)(a-b-1)$



$4. \ \ a^2b^2+1-a^2-b^2 \\ \\ \Rightarrow\ 1-a^2-b^2+a^2b^2 \\ \\ \Rightarrow\ (1-a^2)-b^2(1-a^2) \\ \\ \Rightarrow\ (1-a^2)(1-b^2) \\ \\ \Rightarrow\ (1+a)(1-a)(1+b)(1-b) \\ \\ \\ OR\\ \\ \\ \Rightarrow\ a^2b^2+1-a^2-b^2 \\ \\ \Rightarrow\ a^2b^2-a^2-b^2+1 \\ \\ \Rightarrow\ a^2(b^2-1)-(b^2-1) \\ \\ \Rightarrow\ (a^2-1)(b^2-1) \\ \\ \Rightarrow\ (a+1)(a-1)(b+1)(b-1)$

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